Robust optimal asset–liability management with delay and ambiguity aversion in a jump-diffusion market

In this paper, we consider a robust optimal asset–liability management problem with delay for an ambiguity-averse investor (AAI), who does not hold a firm belief towards not only the drift parameters but also the jump intensity in the wealth process. The objective of AAI is to maximise the robust value involving the expected utility of a weighted sum of terminal wealth and a penalisation of model uncertainty. In addition, the wealth process is modelled by a stochastic delay equation with jump via introducing the performance-related capital inflow/outflow. Applying the technique of robust optimal theory and Hamilton–Jacobi–Bellman equation, we derive the optimal results and the proof of verification theorem for the general utility functions and liability processes. In the case of the exponential utility function and a positive liability process, we prove the existence and uniqueness of the robust optimal investment strategy and find it naturally satisfies the constraint of no-short selling. Meanwhile, we adopt a special method to solve the corresponding Riccati equation and obtain the value function in explicit form. Furthermore, the results under the complete memory are discussed in detail. Finally, some numerical examples are provided to illustrate the influence of model parameters on the optimal strategy as well as the economic interpretation behind it.